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\begin{document}

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%                      TITLE PAGE + ABSTRACT
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%\wideabs{
\title{
\null
\vskip-6pt \hfill {\rm\small MCTP-02-200} \\
\vskip-9pt~\\
Generalized Cardassian Expansion:\\ Models in Which the Universe is Flat,
Matter Dominated,\\ and Accelerating}

\vspace{.5in}

\author{Katherine Freese}
%\vspace{.5in}

\address{
\vspace{.7cm}
Michigan Center for Theoretical Physics,
%\\  Physics Department,
University of Michigan,
Ann Arbor, MI 48109, USA\\
%$^2$ CERN Theory Division, CH-1211 Geneva 23, Switzerland\\
}
\maketitle


\begin{abstract}

The Cardassian universe is a proposed 
modification to the Friedmann Robertson Walker (FRW) equation in
which the universe is flat, matter dominated, and accelerating.
Here we generalize the original Cardassian proposal to include
additional variants on the FRW equation.  Specific examples
are presented.

In the ordinary FRW equation, the right hand side 
is a linear function of the energy density, $H^2 \sim \rho$.
Here, instead, the right hand side of the FRW equation 
is a different function of the energy density, $H^2 \sim g(\rho)$.
This function returns to ordinary FRW at early times, but
modifies the expansion at a late epoch of the universe.
The only ingredients in this universe are matter and radiation:
in particular, there is {\it no} vacuum contribution.
Currently the modification of the FRW equation is such that the
universe accelerates.
The universe can be flat and yet consist of only matter and
radiation, and still be compatible with observations.  The energy
density required to close the universe is much smaller than in a
standard cosmology, so that matter can be sufficient to provide a flat
geometry.  The modifications may arise, e.g., as a consequence of
our observable universe living as a 3-dimensional brane in a higher
dimensional universe.  The Cardassian model survives several
observational tests, including the cosmic background radiation, the age
of the universe, the cluster baryon fraction, and structure formation.
As will be shown in future work,
the predictions for observational tests of the generalized 
Cardassian models
can be very different from generic quintessence models, whether
the equation of state is constant or time dependent.


\end{abstract}
\pacs{}

%}
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%                     BEGINNING OF TEXT
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Recent observations of Type IA Supernovae \cite{SN1,SN2} as well as
concordance with other observations (including the microwave background
and galaxy power spectra) indicate that the universe is accelerating.
Many authors have explored a cosmological constant, a decaying vacuum
energy \cite{fafm,frieman}, quintessence
\cite{stein,caldwell,huey}, and gravitational
leakage into extra dimensions \cite{ddg} as possible explanations for such
an acceleration.

Recently we proposed Cardassian expansion \cite{freeselewis}
(hereafter Paper I)\footnote{The name Cardassian
refers to a humanoid race in Star Trek whose goal is to
accelerate expansion of their evil empire.  This race looks foreign to us
and yet is made entirely of matter.}
as an explanation for acceleration 
which invokes no vacuum energy whatsoever.
In our model the universe is flat and accelerating,
and yet consists only of matter and radiation.
Previously we considered the addition of  a new term to the right
hand side of the FRW equation:
\begin{equation}
\label{eq:new}
H^2 = A \rho + B \rho^n 
\end{equation}
where energy density $\rho$ contains
only matter and radiation (no vacuum) and $n$ is a time-independent 
number with
\begin{equation}
n<2/3 .  
\end{equation}
Here
$H = \dot R / R$ is the Hubble constant (as a function of time) and
$R$ is the scale factor of the universe.  
In the usual FRW equation
$B=0$.  To be consistent with the usual FRW result, we take $A={8\pi
\over 3 m_{pl}^2}$. 
The new term is initially
negligible, and only comes to dominate at redshift $z_{car} \sim {\cal O}(1)$
when $A \rho(z_{car}) = B \rho^n(z_{car})$.
Once it dominates, it causes the universe to accelerate,
as discussed further below.

{\bf Generalized Cardassian Models}

Here we wish to generalize this proposal to other functions
on the right hand side of the FRW equation.
Pure matter (or radiation) alone can drive an accelerated
expansion if the Friedmann Robertson Walker (FRW) equation is modified
to become
\begin{equation}
\label{eq:general}
H^2 = g(\rho) ,
\end{equation}
We take $g(\rho)$ to be
a function of $\rho$ that returns simply to $\rho$ at early 
epochs, but that can drive an accelerated expansion in the recent past
of the universe, after redshifts $z \sim 1$.  
We take the usual energy conservation:
\begin{equation}
\label{eq:energy}
\dot \rho + 3H (\rho + p) = 0 ,
\end{equation}
which gives the evolution of matter:
\begin{equation}
\label{eq:matter}
\rho_M = \rho_{M,0}(R/R_0)^{-3} .
\end{equation}
Here subscript $0$ refers
to today.  Eqs.(\ref{eq:general}) and (\ref{eq:energy})
contain the complete information of 
 the two Friedmann equations.
 
We note here that the geometry is flat,
as required by measurements of the cosmic background radiation
\cite{boom}, since there are no curvature terms in the equation.
Note also that there is no vacuum term in the equation.
This paper does not address the cosmological constant ($\Lambda$) problem;
we simply set $\Lambda=0$.

The simplest example
of the type of behavior in Eq.(\ref{eq:general})
is the sum of two terms:
\begin{equation}
H^2 = \rho + f(\rho)
\end{equation}
where $f(\rho)$ is a different function of $\rho$.  


As mentioned above,
in Paper I, the specific form of $f(\rho)$ that we considered was
$H^2 = A \rho + B \rho^n$ with $n<2/3$ and $n$ constant in time..
Another way to write this equation is
\begin{equation}
\label{eq:new2}
H^2 = A \rho [1 + ({\rho \over \rho_{car}})^{n-1}] .
\end{equation}
The first term inside the bracket dominates initially
but the second term takes over once the energy density
has dropped to the value $\rho_{car}$.
Here, $\rho_{car}$ is the energy density at which the
two terms are equal: the
ordinary energy density term on the right hand side of the FRW
equation is equal in magnitude to the new term.
Hence there are two parameters in the model: one can take them
to be $B$ and $n$,
or equivalently, $\rho_{car}$ and $n$, or equivalently,
$z_{car}$ and $n$.

The new term in the equation (the second term on the right
hand side) is initially negligible.  It only
comes to dominate recently, at the redshift $z_{car} \sim {\cal O}$(1) 
indicated by the supernovae observations.  Once the second term dominates,
it causes the universe to accelerate.  
When the new term is so large
that the ordinary first term can be neglected, we find
\begin{equation}
R \propto t^{2 \over 3n}
\end{equation}
so that the expansion is superluminal (accelerated) for $n<2/3$.
As examples, for $n=2/3$ we have $R \sim t$;
for $n=1/3$ we have $R \sim t^2$; and for $n=1/6$ we have $R \sim t^4$.
The case of $n=2/3$ produces a term in the FRW
equation $H^2 \propto R^{-2}$; such a term looks similar to a curvature
term but is generated here by matter in a universe with a flat geometry.
Note that for $n=1/3$ the acceleration is constant, for $n>1/3$ the
acceleration is diminishing in time, while for $n<1/3$ the acceleration
is increasing (the cosmic jerk).

Note that the parameter $B$ here is chosen to make the second term kick in
at the right time to explain the observations.  As yet we have
no explanation of
the coincidence problem; i.e., we have no explanation
for the timing of $z_{car}$.  Such an explanation would
arise if we had a reason for the required mass scale of $B$,
such as may arise in the context of extra dimensions.


{\bf  Examples of Alternative FRW Equations}

We wish to mention here some alternative forms of $g(\rho)$
in Eq.(\ref{eq:general}).
Wang, Freese, Frieman, and Gondolo \cite{wang}
are studying three Cardassian alternatives:

\noindent
1)  A simple generalization of Eq.(\ref{eq:new}) is:
\begin{equation}
\label{eq:overq}
H^2 = A \rho [1+ ({\rho/\rho_{car}})^{q(n-1)}]^{1/q} .
\end{equation}
Here, $q>0$.  As before, we require $n<2/3$
and $\rho_{car}$ is the energy density at which
the two terms inside the bracket are equal.
The right hand side  returns to $A \rho$ (the ordinary
FRW equation) at early times, but becomes
$\rho^n$ at late times, just as in Eq.(\ref{eq:new2}).
Hence the first term inside the bracket dominates initially
but the second term takes over once the energy density
has dropped to the value $\rho_{car}$.
However, the cross over time period during which the
two terms are roughly comparable is different here.

\noindent
2) Another possibility is
\begin{equation}
\label{eq:poly}
H^2 = D [1 + (\rho/\rho_{car})^q]^{1/q} .
\end{equation}
This example can have a particularly interesting
equation of state.  Gondolo and Freese \cite{gondolo}
are considering treating the right hand side of
Eq.(\ref{eq:poly}) as a single fluid. Then this fluid
behaves as a polytrope of negative index:
\begin{equation}
p \propto - ({\rho \over \rho_{car}})^{1-q},
\end{equation}
which corresponds to a polytrope
$p= K \rho^{1+1/N}$ with negative index $N=-1/q$ and
negative pressure ($K<0$).

\noindent
3) A third possibility modifies the simplest Cardassian
proposal with a logarithm:
\begin{equation}
\label{eq:log}
H^2 = A \rho + B \rho^n {\rm log}^q \rho .
\end{equation}
Many other possibilities for the function $g(\rho)$ in Eq.(\ref{eq:general})
exist.

As will be shown in future work \cite{wang},
the predictions for observational tests of these models
can be very different from generic quintessence models, whether
the equation of state is constant or time dependent.

{\bf The simple Cardassian Model: FRW with additional $\rho^n$ term}

For the rest of this presentation, we study specifically the case
where $g(\rho) = A \rho + B \rho^n$ for constant $n$.
This is the case that was studied in Paper I.
We use it to illustrate the basic properties of a Cardassian model.
First we discuss the phenomenology of this 
and then turn to a discussion of the origin of
this equation\footnote{As discussed below, we were 
motivated to study an equation of this form
by work of Chung and Freese \cite{cf} who showed that terms
of the form $\rho^n$ can generically appear in the FRW equation
as a consequence of embedding our observable universe as a brane
in extra dimensions.}.  

{\it What is the Current Energy Density of the Universe?}

Observations of the cosmic background radiation show that the geometry
of the universe is flat with $\Omega_0=1$.  
In the Cardassian model we need to revisit
the question of what value of energy density today,
$\rho_0$, corresponds to a flat geometry.
We will show that the energy density required to close
the universe is much smaller than in a standard cosmology,
so that matter can be sufficient to provide
a flat geometry.

>From evaluating Eq.(\ref{eq:new}) today, we have
\begin{equation}
\label{hubbletoday}
H_0^2 = A \rho_0 + B \rho_0^n
\end{equation}
The energy density $\rho_0$ that satisfies Eq.(\ref{hubbletoday})
is, by definition, the critical density.
Defining $\rho_0 = \Omega_0 \rho_c$
we find that the critical density $\rho_c$ has been modified from
its usual value, i.e., the number has changed.
We find
\begin{equation}
\rho_c = \rho_{c,old} \times F(n)
\end{equation}
where
\begin{equation}
\label{eq:F}
F(n) = [1+(1+z_{car})^{3(1-n)}]^{-1}
\end{equation}
and
\begin{equation}
\rho_{c,old} = 1.88 \times 10^{-29} h_0^2 {\rm gm/cm^{-3}}
\end{equation}
and $h_0$ is the Hubble constant today in units of 100 km/s/Mpc.
In Figure 1, we have plotted the new critical density $\rho_c$
as a function of the two parameters $n$ and $z_{car}$.

In the (simplest) Cardassian model with new term $\rho^n$,
we see that the value of the critical density can be much lower
than previously estimated.
Since $\Omega_0=1$, we have today's energy density as
$\rho_0=\rho_c$ as given above \footnote{An alternate possible definition
would be to keep the standard value of $\rho_c$ and discuss the
contribution to it from the two terms on the right hand side
of the modified FRW equation.  Then there would be a contribution
to $\Omega$ from the $\rho$ term and another contribution from
the $\rho^n$ term, with the two terms adding to 1.  This is the
approach taken when one discusses a cosmological constant in lieu
of our second term. However, the situation here is different in
that we have only matter in the equation.  The disadvantage of this
second choice of definitions would be that a value of the energy density today
equal to $\rho_c$ according to this second definition would not correspond
to a flat geometry.}.
Note the amusing result that for $z_{car}=2$ and
$n=1/12$, we have $\rho_c = 0.046 \rho_{c,old}$ so that baryons would
close the universe (not a universe we advocate).


\begin{figure}[ht]
\centering
\includegraphics{fplot.eps}
\caption{The ratio $F(n,z_{car})= \rho_c/\rho_{c,old}$ as given by
Eq.(\ref{eq:F}) in the simplest Cardassian model
with new term $\rho^n$. The contour labeled 0.3 corresponds to parameters $n$
and $z_{car}$ roughly consistent with present observations. }
\end{figure}

For the past ten years, a multitude of observations has pointed
towards a value of the matter density $\rho_o \sim 0.3 \rho_{c,old}$. 
The cluster baryon fraction \cite{white,evrard} 
as well as the observed galaxy power spectrum are best fit
if the matter density is 0.3 of the old critical density.
Recent results from the CMB \cite{boom,dasi} are consistent with this value.
In the standard cosmology this result implied that matter could
not provide the entire closure density.  Here, on the other hand,
 the value of the critical density can be much lower
than previously estimated.
Hence the cluster motivated value for $\rho_o$ is now
{\it compatible} with a closure
density of matter, $\Omega_o =1$, all in the form of matter.

For example, if $n= 0.6$ with $z_{car} =1$,
or if $n=0.2$ with $z_{car} = 0.4$, a critical density of matter
corresponds to  $\rho_o \sim 0.3 \rho_{c,old}$, as required
by the cluster baryon fraction and other data. In Figure 1, one can see which
combination of values of $n$ and $z_{car}$ produce the
required factor of 0.3. 
If we assume that the value $\rho_o = 0.3
\rho_{c,old}$ is correct, for a given value of $n$
(that is constant in time) we can compute the value of $z_{car}$ for our
model from Eq.(\ref{eq:F}).  Table I lists these values of $n$ and
$z_{car}$.  Henceforth, we shall focus on these combinations
of parameters.

{\it Age of the Universe}

In the (simplest) Cardassian model with new term $\rho^n$, 
the universe is older due to the presence of
the second term.   In Table I, we present the age of the universe
for various values of $n$ (under the assumption that
$\rho_0=0.3 \rho_{c,old}$).

\begin{table}
\begin{center}
\begin{tabular} {ccc}
$n$ & $z_{car}$ & $H_0 t_0$ \\ \hline
0.60 & 1.00 & 0.73 \\
0.50 & 0.76 & 0.78 \\
0.40 & 0.60 & 0.83 \\
0.30 & 0.50 & 0.87 \\
0.20 & 0.42 & 0.92 \\
0.10 & 0.37 & 0.95 \\
0.00 & 0.33 & 0.99 \\
\end{tabular}
\end{center}
\caption{Values of ${z_{car}}$ for various values of $n$ corresponding
to a universe with $\rho_0 = 0.3\rho_{c,old}$.  The age of the
universe today $t_0$ corresponding to the two parameters $n$ and
$z_{car}$ is listed in the last column, where $H_0$ is the value of the
Hubble constant today.  }
\end{table}

If one takes $t_0 > 10$Gyr as the lower bound on globular cluster ages,
then one requires $t_0 H_0 > 0.66$ for $h_0=0.65$.
If one requires globular cluster ages greater than 11 Gyr
\cite{kc}, then $t_0 H_0 > 0.73$ for $h_0=0.65$.
All values in Table I satisfy these bounds.

{\it Structure Formation}

In the simplest Cardassian model,
the new term $\rho^n$ becomes important only at $z_{car} \sim 1$;
thus early structure formation is not affected.  Below we discuss
the impact on late structure formation during the era
where the Cardassian term is important. This term accelerates
the expansion of the universe, and freezes out perturbation growth
once it dominates (much like when a curvature term dominates); this
freezeout happens late enough that it is relatively unimportant.
To obtain an idea of the type of effects that we may find, instead
of analyzing the exact perturbation equations with metric perturbations
included, we will merely modify the time dependence of the scale
factor in the usual Jeans equation.
For now we take the standard equation for perturbation growth;
as a caveat, we warn that recent structure formation may be further
modified due to a change in Poisson's equations as described below.
For we now we take
\begin{equation}
\label{eq:struct}
\ddot{\delta} + 2(\dot R / R) \dot \delta = 4 \pi \rho \delta/m_{pl}^2
\end{equation}
where $\delta = (\rho - \bar \rho)/\bar\rho$ is the fluid overdensity.
Now one must substitute Eq.(\ref{eq:new}) for $\dot R/R$.
In the standard FRW cosmology with matter domination, $R \sim t^{2/3}$,
and there is one growing solution to $\delta$ with
$\delta \sim R \sim t^{2/3}$.  This standard result still applies
throughout most of the (matter dominated) history of the universe
in our new model, so that structure forms in the usual way.

Modifications set in once the new Cardassian term becomes important.
When $R \sim t^p$, Eq.(\ref{eq:struct}) can be written

\begin{equation}
\label{eq:struct2}
\delta ''(x) + {2 p \over x} \delta '(x) - {3 p^2 \over 2} x^{-3p}
\delta =0 ,
\end{equation}


\noindent where $x \equiv t/t_0$ with $t_0$ denoting the time today and
superscript prime refers to $d/dx$.  This equation can generally be
solved in terms of Bessel functions for constant $p$ (such as is the
case once the Cardassian term completely overrides the old term). A
simple example is the case of $n=2/3$ and $p=1$; in the limit $x>>3/4$,
the last term in Eq.(\ref{eq:struct2}) can be dropped and the equation
is solved as $\delta(t) = a_1 + a_2 t^{-1}$.  Perturbations cease
growing and become frozen in. This result agrees with the expectation
that in a universe that is expanding more rapidly, the overdensity will
grow more slowly with the scale factor.  As mentioned at the outset, as
long as the Cardassian term becomes important only very late in the
history of the universe, much of the structure we see will have already
formed and be unaffected.  

{\it Doppler Peak in Cosmic Background Radiation}

Here we argue that the location of the first Doppler peak is only mildly
affected by the new Cardassian cosmology. We evaluate the shift
explicitly for the case of an additional $\rho^n$ term.
We need to calculate the
angle subtended by the sound horizon at recombination.  In the standard
FRW cosmology with flat geometry, this value
corresponds to a spherical harmonic with $l=200$. A peak at this
angular scale has indeed been confirmed \cite{boom}.  In the Cardassian
cosmology we still have a flat geometry. Hence, we can still write
\begin{equation}
\label{eq:theta}
\theta = s_*/d,
\end{equation}
 where $s_*$ is the sound horizon
at the time of recombination $t_r$ and ${d}$ is the distance
a light ray travels from recombination to today.  To calculate
these lengths, we use the fact that for a light ray $ds^2=0=
-dt^2 + a^2 d\vec{x}^2$ to write
\begin{equation}
\label{eq:light}
d = \int_{t_r}^{t_0} dt/a .
\end{equation}
Following the notation of Peebles \cite{peebles},
we define the redshift dependence of $H$ as
\begin{equation}
H(z) = H_0 E(z)
\end{equation}
so that Eq.(\ref{eq:light}) can be written
\begin{equation}
\label{eq:path}
d = {1 \over H_0 R_0} \int_{0}^{z_r} {dz \over E(z)} .
\end{equation}
Similarly, the sound horizon at recombination is
\begin{equation}
\label{eq:sound}
s_* = \int_{z_r}^\infty dt/a .
\end{equation}

In standard matter dominated FRW cosmology with $\Omega_{m,0} = 1$,
$E(z) = (1+z)^{3/2}$ in Eq.(\ref{eq:path}) and $d = 2/H_0 R_0$.

For the cosmology of Eq.(\ref{eq:new}), we have
\begin{equation}
\label{eq:newE}
E(z)^2 = F\times (1+z)^3 + (1-F) \times (1+z)^{3n}
\end{equation}

\noindent with $F$ given in Eq.(\ref{eq:F}).  
As discussed previously, as our standard case we will take
$F \equiv \rho_c / \rho_{c,old} = 0.3$.  We use $h=0.7$ and
$\Omega_b=0.04$. 
The angle subtended by the
surface of last scattering decreases and the location ($l$) of the first
Doppler peak increases by roughly a factor of

\begin{equation}
\mbox{(1.02, 1.11, 1.12, 1.13)} \,\,\, {\rm for} \,\,\,
$n$ =\mbox{ (0.6, 0.3, 0.2, 0.1)} \,\,\, {\rm respectively}
\end{equation}

\noindent compared to the usual FRW universe with $\rho_{o} = \rho_{c,old}$. 
This shift still lies within
the experimental uncertainty on measurements of the location of the
Doppler peak.

We note the following: in the same way that a nonzero $\Lambda$ may make
the current CBR observations compatible with a small but nonzero
curvature of the universe, similarly 
a nonzero Cardassian term could also allow for a
nonzero curvature in the data.  A more accurate study of the effects of
Cardassian expansion on the cosmic background radiation (including the
first and higher peaks) is the subject of a future study.

{\it The Cutoff Energy Density}

As mentioned previously, another way 
to write Eq.(\ref{eq:new}) is

\begin{equation}
\label{eq:alternate}
H^2 = A \rho \bigl[1+({\rho / \rho_{car}})^{n-1}\bigr]
\end{equation}

\noindent where $\rho_{car} \equiv \rho(z_{car})= A/B$ (see Eq.(\ref{eq:new2}).
This notation
offers a new interpretation; it indicates that the second term only
becomes important once the energy density of the universe drops below
$\rho_{car}$, which has a value a few times the critical
density. Hence, regions of the universe where the density exceeds this
cutoff density will not experience effects associated with the
Cardassian term.  In particular, we can be reassured that the new term
won't affect gravity on the Earth or in the Solar System.  The density of
water on the Earth is 1 gm/cm$^3$, which is 28 orders of magnitude
higher than the cutoff density.

{\it Comparing $\rho^n$ Cardassian to Quintessence}

We note that, with regard to observational tests, one can make a
correspondence between the $\rho^n$ Cardassian and Quintessence models
for time independent $n$; we
stress, however, that the two models are entirely
different. Quintessence requires a dark energy component with a
specific equation of state ($p = w\rho$), whereas the only ingredients
in the Cardassian model are ordinary matter ($p = 0$) and radiation
($p = 1/3$). However, as far as any observation that involves only
$R(t)$, or equivalently $H(z)$, the two models predict the same
effects on the observation.  Regarding such observations, we can make
the following identifications between the Cardassian and quintessence
models: $n \Rightarrow w+1$, $F\Rightarrow \Omega_m$, and $1-F
\Rightarrow \Omega_Q$, where $w$ is the quintessence equation of state
parameter, $\Omega_m= \rho_m/\rho_{c,old}$ is the ratio of matter
density to the (old) critical density in the standard FRW cosmology
appropriate to quintessence, $\Omega_Q= \rho_Q/\rho_{c,old}$ is the
ratio of quintessence energy density to the (old) critical density,
and F is given by Eq.(\ref{eq:F}).  In this way, the Cardassian model
can make contact with quintessence with regard to observational tests.

All observational constraints on quintessence that depend only on the
scale factor, $R(t)$ (or, equivalently, $H(z)$) can also be used to
constrain the Cardassian model.  However, because the Cardassian model
requires modified Einstein equations (see below), the gravitational
Poisson's equations and consequently late-time structure formation may
be changed; e.g., the redshift dependence of cluster abundance should
be different in the two models.  These effects (and others, such as
the fact that quintessence clumps) may serve to distinguish the
Cardassian and quintessence models. 

Although the $\rho^n$ Cardassian and quintessence models
are indistinguishable as far as any observational tests
involving only $R(t)$,  generalized Cardassian models
can be completely different from generic quintessence models, even as
far as predictions for supernova data.
A future paper \cite{wang} will demonstrate the differences in the data
between generalized Cardassian models discussed above
in Eqs.(\ref{eq:overq}), (\ref{eq:poly}), and (\ref{eq:log})
and generic quintessence models.

{\it Best Fit of Parameters to Current Data for $\rho^n$ Cardassian}

We can find the best fit of the Cardassian parameters $n$ and $z_{car}$
to current CMB and Supernova data.  The current best fit is obtained for
$\rho_o = 0.3 \rho_{c,old}$ (as we have discussed above) and $n<0.4$
(equivalently, $w < -0.6$) \cite{bean,hm}.  
In Table I one can see the values of
$z_{car}$ compatible with this bound, as well as the resultant age of the
universe.  
As an example, for $n= 0.2$ (equivalently, $w=-0.8$), 
we find that $z_{car} = 0.42$.
Then the position of the first Doppler peak is shifted by a factor of
1.12.  The age of the universe is 13 Gyr.  The cutoff energy density is
$\rho_{car} = 2.7 \rho_c$, so that the new term is important only for
$\rho < \rho_{car} = 2.7\rho_c$.  Hence, as mentioned above,
the $\rho^n$ Cardassian term won't affect the physics on the Earth or in the
solar system in any way.

We note the enormous uncertainty in the current data; future
experiments (such as SNAP \cite{snap}) will constrain these parameters
further.

{\it Extra Dimensions}

A $\rho^n$ Cardassian term may arise as a consequence of embedding our
observable universe as a 3+1 dimensional brane in extra dimensions.
Chung and Freese \cite{cf} showed that, in a 5-dimensional universe
with metric
\begin{equation}
\label{eq:metric}
ds^2 = -q^2(\tau,u) d\tau^2 + a^2(\tau,u) d{\vec{x}}^2 + b^2(\tau,u)du^2 ,
\end{equation}
where $u$ is the coordinate of the fifth dimension,
one may obtain a modified FRW equation on our observable
brane with $H^2 \sim \rho^n$ for any $n$ (see also \cite{binetruy}).
This result was obtained
with 5-dimensional Einstein equations plus the Israel boundary
conditions relating the energy-momentum on our brane to the
derivatives of the metric in the bulk.

We do not yet have a fundamental higher dimensional theory, i.e.,
a higher dimensional $T_{\mu\nu}$,
which we believe describes our universe.
Once we have this, we can write down the modified four-dimensional
Einstein's equations and compute the modified Poisson's equations,
as would be required, e.g., to fully understand latetime structure formation.

There is no unique 5-dimensional energy momentum tensor
$T_{\mu\nu}$ that gives rise to Eq.(\ref{eq:new}) on our brane.
Hence, in \cite{freeselewis} we constructed
an example which is easy to find
but is clearly not our universe, simply as a proof that such
an example can be written down.
Following \cite{cf} (see Eqs. (24) and (25) there with $F(u) = u$),
we have constructed an example of a bulk
$T_{\mu\nu}$ for arbitrary $n$ in $H^2 \sim \rho^n$, matter
on the brane as in Eq.(\ref{eq:matter}), and with $q = b$ in
Eq.(\ref{eq:metric}). We display only $T_0^0$ here:
\begin{equation}
\label{eq:t00}
{\kappa_5^2 T_0^0 = -\frac{ 3^{-\frac{4+n}{n}} B^{-\frac{2}{n}}\epsilon 
}{n^2\tau^2}
\left[ 4 \cdot 81^{\frac{1}{n}} B^{\frac{2}{n}} - 16^{\frac{1}{n}} \kappa^4_5
(\frac{1}{n\tau})^{\frac{4}{n}} (n^2\tau^2+u^2)\right]}
\end{equation}
where
\begin{equation}
{\epsilon =
\exp\left[-(2/3)^{\frac{2+n}{n}}B^{-\frac{1}{n}}\kappa^2_5
(\frac{1}{n\tau})^{\frac{2}{n}}u\right]}
\end{equation}
and the constant $\kappa_5$ is related to the 5-dimensional Newton's
constant $G_5$ and 5-D reduced Planck mass $M_5$ by the relation
$\kappa_5^2 = 8 \pi G_5 = M_5^{-3}$.
This is merely one (inelegant) example of many bulk $T_{\mu\nu}$ that
produce Cardassian expansion.
We hope a more elegant $T_{\mu\nu}$ may be found, perhaps
with a motivation for the required value of $B$.

{\bf Conclusion}

We have proposed $H^2 \sim g(\rho)$ as a modification to the FRW
equations in order to suggest an explanation of the recent acceleration
of the universe.  In the Cardassian model, the universe can be flat and
yet matter dominated.  We have found that the Cardassian modifications
can become important after $z_{car} = \mathcal{O}$$(1)$
and can drive acceleration of the universe.  
We have found that matter alone can be
responsible for this behavior.  The current
value of the energy density of the universe is then smaller than in the
standard model and yet is at the critical value for a flat geometry.
Structure formation is unaffected before $z_{car}$.  The age of the
universe is somewhat longer.  The first Doppler peak of the cosmic
background radiation anisotropy is shifted only slightly and remains consistent
with experimental results.  Such a modified FRW equation may result from
the existence of extra dimensions.  Further work is required to find a
simple fundamental theory responsible for Eq.({\ref{eq:new}).
In this presentation, generalized Cardassian models were discussed.
As will be shown in future work \cite{wang},
the predictions for observational tests of these models
can be very different from generic quintessence models, whether
the equation of state is constant or time dependent.

\acknowledgments

This paper reflects work with collaborators Matt Lewis, Paolo Gondolo,
Yun Wang, and Josh Frieman.  K.F. also thanks Ted Baltz, Daniel Chung,
Richard Easther, Gus Evrard, Wayne Hu, Lam Hui, Will Kinney, Risa
Wechsler, and especially Jim Liu for many useful conversations and
helpful suggestions.  K.F. acknowledges support from the Department of
Energy via the University of Michigan, and thanks the Kavli Institute
for Theoretical Physics at the University of California, Santa
Barbara for hospitality. 
This research was supported in part by the National Science
Foundation under Grant No.\ PHY99-07949.


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%                            REFERENCES
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